Locally Contractive Iterated Function Systems

Abstract

An iterated function system on $\mathscr{X}\subset\mathbb{R}^d$ is defined by successively applying an i.i.d.sequence of random Lipschitz functions from to $\mathscr{X}$ to $\mathscr{X}$. This paper shows how $F _n = f_1\circ\ldots\circ f_n$ may converge even in the absence of the strong contraction conditions, for instance, Lipschitz constant smaller than 1 on average,which earlier work has required. Instead, it is posited that there be a region of contraction which compensates for the noncontractive or even expansive part of the functions. Applications to queues, to self-modifying random walks and to random logistic maps are given.